Radosław Opoka | Searcharxiv

arXiv2026

Dimension of the accumulation set of any hair for the exponential map

We study the dynamics of the exponential map on the complex plane. The set $Λ_{\mathbf{c}}$ of all points sharing a given itinerary $\mathbf{c}$ is non-empty if and only if $\mathbf{c}$ is an exponentially bounded itinerary. For such itineraries, $Λ_{\mathbf{c}}$ also contains a curve of escaping points, and hence its Hausdorff dimension is at least~$1$. We prove that for every exponentially bounded itinerary this dimension is in fact equal to~$1$. In comparison, for certain itineraries, the set $Λ_{\mathbf{c}}$ exhibits highly complicated topological structures, such as indecomposable continua.

math.DS↗

arXiv2025

Indecomposable continua for unbounded itineraries of exponential maps

We study the dynamics of the exponential maps $E_λ: \mathbb{C} \longrightarrow \mathbb{C}$ defined by $E_λ(z) = λe^z$, where $λ> \frac{1}{e}$. We prove that for itineraries of a certain form, the set of all points sharing the given itinerary, together with the point at infinity, is an indecomposable continuum in the Riemann sphere. These itineraries contain infinitely many blocks of zeros whose lengths increase, and they may be unbounded. We prove that in every such continuum, there exists exactly one point whose $ω$-limit set contains the repelling fixed point of $E_λ$. For every other point, the $ω$-limit set is equal either to the point at infinity, or to the forward orbit of $0$ together with the point at infinity. Thus, we generalize the results of R. Devaney and X. Jarque concerning indecomposable continua for bounded itineraries.

math.DS↗